A Proof of Liouville’s Theorem in complex analysis August 31, 2009

Liouville’s theorem in complex analysis says that the only bounded holomorphic functions are constant. This is my favorite

Proof:

Pretend you had a non-constant, bounded holomorphic function on all of . Since is bounded at , Riemann’s theorem on removable singularities implies that extends to a holomorphic (and hence continuous) function on the Riemann sphere , which is compact. If were not constant, the open mapping theorem would apply to this extension, and the image of the Riemann sphere would be an open subset of . But this cannot be the case, because has no nonempty subsets which are both open and compact.

There are some downsides to this proof. It does not rely on the theory of Riemann surfaces (not a downside). It does, however, rely on some relatively (though not truly) heavy machinery, and in order to be a correct proof, one isn’t allowed to use Liouville’s theorem to develop this machinery (but it is possible). Liouville’s theorem follows rather immediately from Cauchy’s integral formulae, and I don’t personally know how to establish things like analyticity, Riemann’s theorem and the open mapping theorem without this tool (although I would be interested if anyone else does!). I also don’t think it really generalizes very well to other PDE for which a Liouville theorem holds.

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But if I’m not mistaken the whole fact that extends to the point at infinity, which implicitly uses the Riemann removable singularity theorem, relies on the representation of as a power series (which uses the Cauchy formula).

One can prove the removable singularity theorem directly from Cauchy’s integral formula, by showing the formula

— where the above contour integral winds once around a neighborhood of in which remains holomorphic — defines a continuous (and holomorphic) function of z even into the possible singularity. This is the only proof I know, actually, and it doesn’t discuss analyticity, but just like analyticity of holomorphic functions, it follows almost immediately from the representation formula. What would be interesting would be to see a proof of these facts about holomorphic functions without relying on the explicit representation formula, maybe something to do with the notion of capacity in the case of removable singularities.